Roaming Point Perspective: A Dynamic Interpretation of the Visual Refinements of the Greek Doric Temple

Writers, artists, and mathematicians since Vitruvius have attributed the use of the visual refinements as a means by which the Greek builders optically corrected the form of the Doric temple. This study proposes an interpretation in which the visual refinements of the Parthenon are considered from a non-stationary, or “roaming”, point of view. The mathematics of this type of visual space reveals a dynamic zone in which objects visually increase and decrease simultaneously, a behavior consistent with conditions addressed by the visual refinements of the Parthenon.     

The Geometry of the Zodiac

The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial ecliptic. Here wee consider the zodiac as a timepiece or calendar; how this system of planets and constellations emerges from elementary geometric patterns; and how these patterns inform the symbols of the zodiac and frame our world view.     

The Other Mathematical Bridge

This paper contextualises, describes and discusses a student project which takes a particular exploratory approach to using mathematical surface definition as a language and vehicle for co-rational design co-authorship for architecture and engineering. The project has two authors, one from an architectural and one from an engineering educational background. It investigates the metaphorical and operational role of mathematics in the design process and outcomes.     

Construction of invariant ellipsoids in the stabilization problem for a wheeled robot following a curvilinear path

The synthesis control problem for the plane motion of a wheeled robot is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it in the presence of phase and control constraints. For a synthesized control law, invariant ellipsoids—quadratic approximations of the attraction domains of the target trajectory—are constructed, which allow one to check in the course of the robot motion whether the control law can stabilize motion along the current trajectory segment. To take into account constraints on the control, methods of absolute stability theory are applied. The construction of the invariant ellipsoids reduces to solving a system of linear matrix inequalities.     

Optimization of boundary control at one end of a string in the presence of a model nonlocal condition

The paper is devoted to problems of optimal boundary control of string vibrations under nonlocal conditions that relate values of string displacement or its derivative at the boundary point with their values at the interior point. The solutions for all problems of boundary control are obtained in explicit analytic form.     

Identification of frequency of a shifted sinusoidal signal

The article deals with the issue of identification of an unknown frequency of a shifted sinusoidal signal y(t) = σ ₀ + σ sin(ωt + ?). A new approach taken to estimate the frequency of a shifted sinusoidal signal is suggested, which is a robust approach relative to undetermined disturbances present in the measurement of a useful signal. In contrast to known analogs, the given approach permits controlling the time of estimation of the unknown frequency ω. The dimension of the identification algorithm suggested is less than that of known analogs.     

Local and semiglobal stabilization in a cascade with delay

Consideration is given to the problem of stability and stabilization for a system of differential equations with delay of a “cascade” structure. Sufficient conditions for local asymptotic stability as well as “semiglobal” stabilization for a cascade with a linear subsystem are obtained.     

Markov model of a digital stochastic generator

A stochastic process generator based on a chain of modulo-two adders with asynchronous feedback loops is investigated. The generator dynamics is described by the Erlang equations. A mathematical model of the generated stochastic process is proposed. The implementation of the generator using modern electronic devices is considered.     

Systems with a switch and estimation of absolute stability region

Two methods are proposed for estimation of absolute stability region in the space of parameters. The first method uses nontrivial necessary conditions, which follow from stability of switching systems. The second method is a modification of the harmonic balance approach; it is used for approximate determination of existence of periodic modes that arise when stability is lost.     

Criteria of resonance origin in a single-circuit control system with saturation

Studies are made of forced periodic oscillations in a single-circuit control system with a parameter, the dynamics of which is described by resonance equations in the linear approximation. Criteria are suggested of the origin of resonance that is understood as an increase to infinity of the amplitude of forced oscillations in the approximation of the parameter to certain critical values. The principal results relate to the case when answers are defined by bounded nonlinearities of the order of a constant; the summands that decrease to zero are of no importance.     

The iteration method of data censoring in the regression estimation problem

A robust analog of the Nadaraya-Watson regression estimate is considered. A solution is obtained in the class of censor algorithms. A criterion and iteration procedure for determining a censored sample are proposed. The criterion is based on the analysis of residuals (errors) of estimation.     

Robust filtering under nonrandom disturbances: The invariant ellipsoid approach

We present a simple and universal observer-based approach to solving the problem of robust filtering of unknown-but-bounded exogenous disturbances. The heart of this approach is the method of invariant ellipsoids. Application of this technique allows for a reformulation of the original problem in terms of linear matrix inequalities with reduction to semidefinite programming and one-dimensional optimization, which are easy to solve numerically. Continuous-time and discrete-time cases are studied in equal detail. The efficacy of the approach is demonstrated via the double pendulum example.     

Estimation of non-stationary delay for a linear discrete dynamic plant

A nonlinear recurrent gradient-type algorithm is proposed for estimation of nonstationary delay arising in finite impulse response of a linear discrete dynamic plant. Convergence of the algorithm is investigated. Using linear matrix inequalities, its optimal functioning is investigated in order to provide for the maximal rate of convergence of the estimation error to zero.     

Two classes of extreme motion trajectories of a point with varying specific impulse

The variational problem to determine the optimal trajectories of the center of mass of a spacecraft moving in the central Newtonian field with limited power and specific momentum was considered. The extremal trajectory was shown to contain elements of zero, intermediate, and maximal power with a constant or variable specific momentum. For the circular and spiral trajectories, extremal analytical solutions were established. These elements can find use in the problems of departure and transfer to the parking orbit, as well as in the problems of orbital transfers. The problem of transfer to the given elliptical orbit was considered as an example.     

About a class of systems preserving the stability property at negative feedbacks

For linear and nonlinear systems are obtained tests for conserving the property of asymptotic stability at switching arbitrary coordinatewise negative feedbacks.     

Control of single-sector economy over a finite time interval with allowance for employer consumption

The problem of optimal control of a single-sector economics over a finite time interval was studied for the class of the linear uniform production functions in terms of the criterion for maximum employer consumption under exponential growth of the manpower resources. Solution was obtained in the form of the “turnpike theorem.” Its properties were considered. The results obtained were specified for the case of the Cobb-Douglas production function.     

Organization of self-diagnosis of the discrete multicomponent systems a structure like bipartite quasicomplete graphs

Organization of an efficient self-diagnosis of the multicomponent computer and communication systems of diverse structures always attracted attention of the researchers and engineers. A method to solve these problems is presented in the paper by way of the example of a system whose structure is modeled by a uniform ordinary bipartite graph of diameter d = 3, any degree s > 1, and any number n of vertices, where n = s(s ? 1) + 1. The method requires checking of (s ? 1)³ graph loops of length eight each, which is smaller than the number s ²(s ? 1) + s of checks of single graph edges.     

Organization of the remote access to tomographic data through a slow communications link

Consideration is given to the problem of organization of the remote access to tomographic data obtained as a result of the investigation on the computerized tomograph through a slow communications link. The proposed access strategy together with the developed and realized method of tomographic data compression makes it possible to help users (physicians, scientists, and etc.) work with remote tomographic data, e.g., using the Internet. This, in its turn, opens up vast possibilities for using the data of tomographic investigations for conducting medical consultations “at a distance,” videoconferences, organizing network seminars for medical staff training, and etc.